CHAP. I] PERFECT, MULTIPLY PERFECT, AND AMICABLE NTIMBEBS. 15 that he has surpassed all analysis up to the present. Goldbach67 called Euler's attention to these remarks and stated that they were probably taken from Mersenne, the true sense not being followed. Wm. Leybourn68 listed as the first ten perfect numbers and the twentieth those which occur in the table of Bungus.42 "The number 6 hath an eminent Property, for his parts are equal to himself." Samuel Tennulius, in his notes (pp. 130-1) on lamblicus,4 1668, stated that the perfect numbers end alternately in 6 and 8, and included 130816 = 256-511 and 2096128 = 1024-2047 among the perfect numbers. Tassius69 stated that all perfect numbers end in 6 or 8. Any multiple of a perfect or abundant number is abundant, any divisor of a perfect number is deficient. He gave as the first eight known perfect numbers the first eight listed by Mersenne.60 Joh. Wilh. Pauli70 (Philiatrus) noted that if 2n—1 is a prime, n is, but not conversely. For n = 2, 3, 5, 7, 13, 17, 19, 2n-l is a prime; but 2n-l is divisible by 23, 223-l by 47, and 241-1 by 83, the three divisors being 2n+l. G. W. Leibniz71 quoted in 1679 the facts stated by Pauli and set himself the problem to find the basis of these facts. Returning about five years later to the subject of perfect numbers, Leibniz implied incorrectly that 2P — 1 is a prime if and only if p is. Jean Prestet72 (fl690) stated that the fifth,.. ., ninth perfect numbers are 23550336 [for 33...], 8589869056, 137438691328, 238584300813952128 [for 2305... 39952128], 2513 -2256. [Hence 2n~1(2n-l) for n = 13, 17, 19, 31, 257. The numerical errors were noted by E. Lucas,124 p. 784.] Jacques Ozanam73 (1640-1717) stated that there is an infinitude of perfect numbers and that all are given by Euclid's rule, which is to be applied only when the odd factor is a prime. Charles de Neuveglise74 proved that the products 3-4,..., 8-9 of two consecutive numbers are abundant. All multiples of 6 or an abundant number are abundant. "Correspondence Math. Phys., ed., Fuss, 1,1843; letters to Euler, Oct. 7,1752 (p. 584), Nov. 18 (p. 593). ^Arithmetical Recreations; or Enchriridion of Arithmetical Questions both Delightful and Profitable, London, 1667, p. 143. ""Arithmeticae Empiricae Compendium, Johazmis Adolfi Tassii. Ex recensione Henrici Siveri, Hamburgi, 1673, pp. 13, 14. 70De numero perfecto, Leipzig, 1678, Magister-disputation. "Manuscript in the Hannover Library. Cf. D. Mahnke, Bibliotheca Math., (3), 13, 1912-3, 53-4, 260. 72Nouveaux elemens des Mathematiques, ou Principes generaux de toiites les sciences, Paris, 1689, I, 154-5. "Recreations mathematiques et physiques, Paris and Amsterdam, 2 vols., 1696, I, 14, 15. 74Traite" methodique et abrege" de toutes les mathematiques, Trevoux, 1700, tome 2 (L'arith- me"tique ou Science des nombres), 241-8.